A powerful diagnostic tool of analytic solutions of ordinary second-order linear homogeneous differential equations
- Dimitris M Christodoulou^{1}Email author and
- Qutaibeh D Katatbeh^{2}
https://doi.org/10.1186/s13662-017-1272-5
© The Author(s) 2017
Received: 30 March 2017
Accepted: 10 July 2017
Published: 26 July 2017
Abstract
Very few of the ordinary second-order linear homogeneous (OSLH) differential equations are known to possess analytic, closed-form solutions and there is no general theory to predict whether such solutions exist in each particular case. In this work, we present a powerful new method for the discovery of closed-form solutions in the entire class of the OSLH differential equations. A sufficient condition for the existence of such solutions is that the equations can be transformed to forms with constant coefficients and we show that two predictors exist, a nonlinear first-order Bernoulli equation with index \(n=3/2\) and a nonlinear second-order equation. There is no need to solve these equations, only to verify them based on the contents of a given OSLH equation. Because the predictors can be verified quite easily in all cases, we believe that this methodology is an important new diagnostic tool for studying all the equations of mathematical physics that belong to the OSLH class.
Keywords
MSC
1 Introduction
We derive the predictor equations in Section 2, and we apply the method to some well-known equations of applied mathematics in Section 3. Finally, we summarize and discuss our results in Section 4.
2 Discovery of closed-form solutions
2.1 Case 1: \(\widehat{b}=\mathrm{const.}\equiv B\) in equation (2)
For b̂ equal to a constant B, equation (2) has constant coefficients and it can easily be solved analytically [6]. Using \(\widehat{b}=B\) in equation (4), we find the first-order Bernoulli equation (3) that must be satisfied by \(b(x)\) and \(c(x)\). This equation also determines the value of the constant B self-consistently. We solve the Bernoulli equation (3) in Appendix 1, where we obtain generalized forms of the second-order Cauchy-Euler equation [2, 4].
On the other hand, when b̂ is not a constant, the predictor does not detect closed-form solutions and then the predictor of Case 2 below must be applied.
2.2 Case 2: the canonical form of equation (2)
As was described above, for a given OSLH differential equation in canonical form, it is sufficient to test whether its coefficient \(c(x)\) satisfies equation (10). Nevertheless, we may ask whether this equation can be solved analytically, in which case we can obtain the general form of \(c(x)\) for which closed-form solutions are guaranteed to exist. The answer, perhaps surprisingly, is yes. We defer the general study of equation (10) to Appendix 3.
3 Applications of the method
3.1 Cauchy-Euler equation
3.2 Chebyshev equation
3.3 Bessel equations of order N
3.4 Equations transformed to the Bessel type
3.5 CDOS equation
4 Summary and discussion
We have discovered two predictors (expressed by the differential equation (3) and by the combination of equations (4) and (8)) for analytic studies of all OSLH differential equations (Section 2). The latter predictor is also described by equation (9) and it is considerably simplified (see equation (10)) if the given OSLH equation is in canonical form. The first predictor is applied to the general form (1), whereas the second predictor is applied to the canonical form (5) after the first predictor has returned negative results (or to the general form (1) with \(b= 0\) or \(b=2/(x+C)\), where C is a constant). When either predictor is satisfied, the result is positive and the given OSLH equation is guaranteed to admit a transformation to a form with constant coefficients that can easily be solved in closed form.
In Section 3, we have analyzed some differential equations of wide interest in applied mathematics [1–9] and we confirmed that they possess analytic, closed-form solutions. We believe that this new diagnostic tool will prove very useful in future studies of many more OSLH differential equations of interest to theoretical physicists and applied mathematicians.
We note, however, that this method of analysis cannot detect the existence of one analytic particular solution in the cases where the other linearly independent solution cannot be written also in closed form. An example kindly provided by the reviewers is the set of equations \(y^{\prime\prime} \pm x y^{\prime} \mp y = 0\), which possess the particular solution \(y=x\) and another particular solution that cannot be written in closed form (it contains the error function \(\operatorname{erf}(\sqrt{x^{2}/2})\)). In this case, both predictors determine (correctly) that the general solutions of these equations do not have closed forms, but they do not detect the solution \(y=x\).
Furthermore, the transformation (2) is not appropriate and does not work for equations whose canonical form happens to have a constant coefficient. An example is the equation \(y^{\prime\prime} + 2x y^{\prime} + x^{2} y = 0\) whose canonical form is \(u^{\prime \prime} - u = 0\) with \(y(x) = u(x) \exp(-x^{2}/2)\), and that can be solved easily. For this reason, the transformation of the general equation (1) to its canonical form should always be attempted before the above methodology is applied.
Declarations
Acknowledgements
We thank the referees for their suggestions that led to an improved presentation of our results. During this research project, DMC was fully supported by the University of Massachusetts Lowell and QDK was fully supported by the Jordan University of Science and Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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